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Joined 2026-09-21 Message
Incoming links: Loop variant
Algorithm termination 2026-10-18
Algorithm termination means that execution finishes after finitely many steps on the inputs under consideration. Specify that domain: the stalled loop in partial correctness terminates for but runs forever for . A successful run on one input does not establish termination on another.
A useful method is a loop variant that is a nonnegative integer at every condition check and strictly decreases whenever the condition is true and the body completes. Also prove that condition evaluation and each body execution terminate. An infinite run would then give an infinite strictly decreasing sequence of nonnegative integers, contradicting their well-founded relation. This establishes algorithm termination without enumerating all executions.
For the summation loop verification, begins at and decreases by exactly one per iteration. The body executes exactly times, including zero times for . Merely bounding a decreasing measure below is insufficient for real numbers: decreases forever while staying positive. The choice of a well-founded relation supplies the essential progress argument; well-founded progress in recursion applies the same idea to recursive calls.
Coupled binary exponentiation 2026-10-31
To compute two nonnegative integer powers in one while loop, maintain a separate accumulator, current base and remaining exponent for each exponentiation by squaring. The paired loop invariant is and . For each positive remaining exponent, transfer its odd factor to its accumulator and halve the exponent while squaring its base. A zero remaining exponent is left unchanged. The mathematical integer is a loop variant: at least one positive term strictly decreases in every iteration. Both powers are therefore complete when the loop stops. The iteration count is , with constant space and a constant number of multiplications per iteration. This is an exact-arithmetic identity; floating-point rounding introduces a separate error in a numerical implementation.
Repeatedly apply a constraint propagation operation that only deletes values from finite constraint domains, preserves their order, and preserves every valid assignment. The sum of domain sizes is a nonnegative loop variant at every changing step, so repetition reaches a fixed point. For initial candidate values there are at most changing steps. An empty domain proves failure; a non-singleton fixed point may still require backtracking or explicit enumeration. Preservation of solutions, rather than mere deletion, is essential.
Formal methods 2026-11-05
Formal methods use mathematically defined models, specifications and reasoning techniques to describe and analyze systems. They include writing a formal specification, proving properties through formal verification, exploring a finite model by model checking and deriving information through static program analysis. A method's guarantee is relative to its model and assumptions.
For example, a loop summing an array can maintain the invariant that the current accumulator equals the sum of the entries already visited. Initialization, preservation by one iteration and the exit condition establish the returned sum; a separate termination argument shows that an answer is reached. Arithmetic and array bounds must be specified, since a proof using unbounded integers need not describe overflowing machine arithmetic.[ref]
An Axiomatic Basis for Computer Programming by Tony Hoare. assignment and iteration rules, and reservations about arithmetic and termination
Formal verification Created 2026-10-13 Updated 2026-10-18
Establishing specified properties of a system using formal proofs about a precise model. Formal verification connects an implementation to a formal specification; correctness of the model and its assumptions still matters when applying the result to a physical system.
For a small example, the summation loop verification below starts with executable steps and asks two separate questions: does any returned value satisfy the postcondition, and does execution reach that return? A loop invariant answers the first; a loop variant answers the second. This turns an unbounded family of permitted inputs into a few reusable proof obligations.
The model must specify programming language semantics, including the meaning of variable assignments, comparisons and integer arithmetic. A proof about exact mathematical integers need not describe a machine that wraps on integer overflow. Similarly, a correct formal proof of a weak or mistaken formal specification does not establish the intended behavior. Choosing the right formal specification and relating the model to the implemented system remain substantive tasks.
Software testing examines chosen executions and can expose a counterexample. A formal proof instead justifies a statement throughout its stated model; it does not require executing every permitted input. This makes formal verification valuable when rare boundary cases matter, while software testing can still reveal an incorrect model or misunderstood requirement.
In 1969, Tony Hoare presented rules for variable assignment, composition and iteration in a small programming language. His paper explicitly says those rules do not prove successful termination, and discusses the dependence of arithmetic reasoning on the chosen overflow behavior. These reservations already make the model's scope part of the argument.[ref]
An Axiomatic Basis for Computer Programming by Tony Hoare. Hoare, An Axiomatic Basis for Computer Programming, §2 Computer Arithmetic, pp. 576–577; §§3.1–3.4, pp. 577–578; §4 General Reservations, pp. 578–579
Greedy Egyptian fraction expansion 2026-10-25
For a positive proper residual fraction , with , choose , append the unit fraction , and replace the residual by in lowest terms. This is a greedy algorithm: the chosen unit fraction is the largest one not exceeding the residual.
The inequalities imply . Thus the numerator of the reduced residual strictly decreases while positive, proving algorithm termination by a nonnegative-integer loop variant. The loop invariant is that the accumulated sum plus the residual is the original rational number. Since and , the new residual is less than . If it remains positive, its next chosen denominator is therefore strictly greater than ; this expansion also satisfies the distinct-denominator convention. The proof does not bound the denominators by the input's fixed-width range. Arbitrary-precision arithmetic, checked conversion, or a different representable expansion can therefore be necessary in code.
Hoare logic Created 2026-10-18 Updated 2026-11-05
Hoare logic proves properties of programs through Hoare triples . A precondition constrains initial states and a postcondition constrains returned states. The basic rules establish partial correctness; termination requires a further argument.[ref]
For a program expression whose evaluation is defined and has no side effects, the assignment rule isThe substitution replaces the free occurrences of in the desired postcondition. For example, obtaining after
x := x+1 requires beforehand, so suffices. Integer overflow must be excluded or handled by the stated arithmetic model.Sequential composition joins a first postcondition to the next precondition. For a while loop, proving establishesHere is the loop invariant. For
while x > 0 do x := x-1, starting with an exact nonnegative integer, is preserved and the false exit guard yields . The decreasing nonnegative integer is also a loop variant, establishing total correctness for this example.[ref] Loop variant 2026-10-18
A decreasing measure used to prove algorithm termination. A nonnegative integer loop variant must strictly decrease after every completed body whose condition was true; the body itself must terminate.[ref]
Loop invariants. CS 2112 Fall 2019; Loop invariants, Using loop invariants to show code is correct: Termination and the decrementing function
The prefix maximum correctness example uses . It counts unread entries and is independent of whether the new entry changes the maximum. Ties therefore cannot stall progress: increases even when stays the same. An integer bounded below cannot strictly decrease indefinitely, which supplies the missing algorithm termination argument.
Control flow and four proof obligations for the prefix maximum correctness example
. Initialization, preservation, and exit establish partial correctness; the loop variant establishes algorithm termination and hence total correctness. The loop invariant is restored after both variable assignments. The input array stays unchanged throughout. Details.
Use two indices, initially the first and last positions of the ordered sequence. The two elements occupy distinct positions, so continue only while
i < j:i := 0; j := n-1;
WHILE i < j DO
t := S[i] + S[j];
IF t = x THEN RETURN TRUE
ELSIF t < x THEN i := i+1
ELSE j := j-1
END
END;
RETURN FALSEThe loop invariant is: a solution exists in the original input if and only if one exists with both indices in
[i,j]. If S[i]+S[j] < x, then S[i]+S[k] <= S[i]+S[j] < x for every k <= j, so index i cannot belong to a remaining solution. If the sum is too large, S[k]+S[j] >= S[i]+S[j] > x for every k >= i, so index j cannot belong to one. This proves preservation of the loop invariant. Equality gives the desired pair, and an interval with fewer than two positions contains none.Every unsuccessful iteration decreases the loop variant
j-i by one, so there are at most sum comparisons and only constant space is needed. Duplicate values are permitted if they occupy distinct positions. Under exact constant-cost real arithmetic and endpoint access,For nonnegative integer
x and positive integer y, initialize the quotient to zero and subtract one divisor at a time:q := 0; r := x;
WHILE r >= y DO
r := r-y;
q := q+1
ENDThe loop invariant is , , , with
x and y unchanged. It is true initially. A body execution changes to and keeps nonnegative because of the guard. The nonnegative integer loop variant strictly decreases by , so termination follows. At exit the guard supplies ; henceThere are exactly body executions. Under the unit-cost model, the time complexity including initialization and the final test is , which is for integer . This is a bound in the value of x, rather than in its encoded bit length.In binary long division, first double the divisor until it exceeds
x, then halve it successively. Keep p = 2^j and d = y*p, so no exponentiation is needed inside a loop:q := 0; r := x;
j := 0; p := 1; d := y;
WHILE d <= x DO
d := 2*d;
p := 2*p;
j := j+1
END;
WHILE j > 0 DO
d := d DIV 2;
p := p DIV 2;
j := j-1;
IF r >= d THEN
r := r-d;
q := q+p
END
ENDThe first loop invariant is , , , and . Every doubling preserves it. For integer inputs, after at most iterations when ,
d > x; if , this loop has no iterations. The unbounded increase of proves termination. Equivalently, one can use the decreasing nonnegative loop variant while the guard is true.At the top of the second loop the loop invariant isIt holds on entry because
d > x = r. Suppose the old d is and the old p is . After the three halving assignments, and the old remainder satisfies . If , the desired new bound already holds. Otherwise subtracting leaves and adding to q preserves because . The divisions by two are exact: before each halving, j > 0 makes both p and d even.The loop variant
j decreases by one each iteration. At exit , so d = y and the loop invariant gives the required remainder bound. The work per iteration is constant in the unit-cost model, and the two loops have the same number of iterations. For ,The case , including , takes constant time. These are arithmetic-operation counts with sufficient range for the doubled values; arbitrary-precision arithmetic incurs additional bit complexity. If the inputs were arbitrary positive real numbers, the uniform bound would depend on rather than only ; the stated is the conventional integer division interpretation.Replace the recursive transfer in part (b) by simultaneous state updates, using a temporary variable to preserve the old
a:PROCEDURE FibIter(n: INTEGER): INTEGER;
VAR k, a, b, oldA: INTEGER;
BEGIN
k := n; a := 1; b := 0;
WHILE k > 1 DO
oldA := a;
a := a + b;
b := oldA;
k := k - 1
END;
RETURN a
END FibIter;For original input , a loop invariant isIt holds initially because . Each iteration preserves it by the identity from part (b). The loop variant is , a nonnegative integer that decreases by one whenever the guard holds. On exit , so the loop invariant gives , exactly the returned answer. The procedure uses arithmetic operations and stored integer variables, without depending on compiler treatment of tail calls.
Represent the two recursively split arrays by adjacent half-open ranges of the original array. This avoids allocating a fresh pair at every call. The caller supplies a distinct work array with at least as many elements as
a; only that work array is used for merging. The Oberon code isPROCEDURE SortRange(VAR a, work: ARRAY OF INTEGER;
l, r: INTEGER);
VAR m, i, j, k: INTEGER;
BEGIN
IF r-l > 1 THEN
m := l + (r-l) DIV 2;
SortRange(a, work, l, m);
SortRange(a, work, m, r);
i := l; j := m; k := l;
WHILE (i < m) & (j < r) DO
IF a[i] <= a[j] THEN
work[k] := a[i]; i := i+1
ELSE
work[k] := a[j]; j := j+1
END;
k := k+1
END;
WHILE i < m DO
work[k] := a[i]; i := i+1; k := k+1
END;
WHILE j < r DO
work[k] := a[j]; j := j+1; k := k+1
END;
k := l;
WHILE k < r DO
a[k] := work[k]; k := k+1
END
END
END SortRange;
PROCEDURE MergeSort(VAR a, work: ARRAY OF INTEGER);
BEGIN
SortRange(a, work, 0, LEN(a))
END MergeSort;For a range of length , the left length is and the right length is , so as required. Both are positive and smaller than , giving the recursive loop variant analogue for algorithm termination. Empty ranges are also harmless, although the question supplies a nonempty array.
For the first merge loop, regard the two sorted ranges as fixed input sequences and . They are not overwritten during the merge. A loop invariant is that , , , and
work[l:k) is the stable sorted merge of the consumed prefixes a[l:i) and a[m:j). Every emitted element is at most every remaining element. Taking the smaller head preserves this property, and choosing the left head on equality preserves stable sorting. The loop variant isIt is nonnegative and decreases by one per iteration. On exit at least one range is exhausted.For the second loop, the remaining left range is already sorted and any right remainder is absent whenever that loop runs. Appending the next left element preserves the same merge loop invariant; its loop variant is . For the third loop, the left range is exhausted and the right remainder is appended in order; its loop variant is . On completing both loops,
work[l:r) is the complete sorted merge and .In the final copying loop, the loop invariant is that
a[l:k) equals work[l:k), work[l:r) remains the correct sorted answer, and positions outside a[l:r) are unchanged. Its loop variant is . At exit the whole answer has been copied back. By strong induction on range length, the recursive calls sort their halves, and the merge preserves the original multiset. ThusThe work array must not alias a: overwriting the source while merging would invalidate the merge loop invariant.Use an Oberon record containing an active coefficient count and a bounded array of coefficients in ascending power order. A count of zero represents the zero polynomial; a count represents . Trailing zero coefficients are allowed, so the count need not be the degree of a polynomial plus one.
CONST Capacity = 1024;
TYPE Polynomial = RECORD
len: INTEGER;
coeff: ARRAY Capacity OF REAL
END;
PROCEDURE Horner(VAR p: Polynomial; x: REAL): REAL;
VAR i: INTEGER; value: REAL;
BEGIN
i := p.len; value := 0.0;
WHILE i > 0 DO
i := i-1;
value := p.coeff[i] + x*value
END;
RETURN value
END Horner;The representation invariant is , with only indices below
len significant. The parameter is passed by reference to avoid a full record copy; the procedure does not modify it.Let be the original coefficient count. At each guard test the loop invariant isThe empty sum is zero initially. If the old index is , after decrementing and updating,which preserves the loop invariant. The loop variant is , decreasing by one until zero. At exit,This is exactly the right-to-left nesting of Horner's method, matching the supplied right fold. It uses arithmetic operations and additional scalar storage.
REAL gives ordinary floating-point evaluation; the displayed identities specify the ideal arithmetic calculation, with the usual rounding in an actual floating-point implementation.Add polynomials by adding equal-power coefficients, interpreting a missing coefficient as zero. Compute into a local record before assigning the output, so the result may safely be the same actual record as either input.
PROCEDURE Add(VAR p, q, result: Polynomial);
VAR t: Polynomial; i: INTEGER; u, v: REAL;
BEGIN
t.len := p.len;
IF q.len > t.len THEN t.len := q.len END;
i := 0;
WHILE i < t.len DO
u := 0.0; v := 0.0;
IF i < p.len THEN u := p.coeff[i] END;
IF i < q.len THEN v := q.coeff[i] END;
t.coeff[i] := u+v;
i := i+1
END;
result := t
END Add;With , the loop invariant is and, for ,
t.coeff[k] is the sum of the two extended input coefficients at . The loop variant is . The inputs remain unchanged throughout the loop, and at exit every active coefficient is correct. The time complexity is for the active arithmetic; assigning the fixed-capacity record can additionally copy its full capacity.For the product, each pair of input terms contributes . Initialize the active result to zero, then accumulate these pairwise contributions:
PROCEDURE Multiply(VAR p, q, result: Polynomial);
VAR t: Polynomial; i, j, k: INTEGER;
BEGIN
t.len := 0;
IF (p.len > 0) & (q.len > 0) THEN
t.len := p.len + q.len - 1
END;
(* Precondition: t.len <= Capacity. *)
k := 0;
WHILE k < t.len DO
t.coeff[k] := 0.0; k := k+1
END;
IF t.len > 0 THEN
i := 0;
WHILE i < p.len DO
j := 0;
WHILE j < q.len DO
t.coeff[i+j] := t.coeff[i+j]
+ p.coeff[i]*q.coeff[j];
j := j+1
END;
i := i+1
END
END;
result := t
END Multiply;The input representation invariants and the capacity precondition are assumed. No coefficient is accessed when either input is the zero polynomial. The initialization loop invariant says coefficients below are zero; its loop variant is
t.len-k.For positive input lengths , the outer loop invariant says that and each coefficient equalsThe inner loop invariant adds the contributions with and to that sum, while . The update adds exactly the next missing term at index , preserving the inner loop invariant. The loop variants are for the outer loop and for the inner loop. On exit all pairs have contributed, yieldingThe product uses arithmetic operations and a temporary coefficient array of capacity
Capacity. Both operations allow trailing zeros; no extra normalization loop is required. Only the active coefficients are meaningful, so inactive record entries are never read by these operations. As in part (a), floating-point coefficients implement these formulas subject to rounding.Evaluate the mixed radix representation from its most significant end, using Horner's method. There is no need to compute the positional weights separately.The digit conditions are additional preconditions. The loop invariant at the loop test isAn empty sum is zero and an empty product is one, so initialization at is correct. For the next digit, writing giveswhich preserves the loop invariant. The loop variant is . At termination , givingThe code executes exactly multiplications, meeting the bound, and uses auxiliary space. When it returns zero. For the time representation in the source, it computes .
def digsToInt(xs: Array[Int], bs: Array[Int]): Int = {
require(xs.length == bs.length)
val N = xs.length
var i = N
var acc = 0
while (i > 0) {
i -= 1
acc = xs(i) + bs(i) * acc
}
acc
}A suitable precondition, including the empty array case, isA base of one is allowed: its only valid digit is zero. The empty product is one, so permits exactly . Each Euclidean division extracts the next least significant digit.The expensive precondition is a caller obligation, rather than a separately computed product inside the conversion. Define . The loop invariant istogether with for each processed digit. The Euclidean division identity proves preservation of the reconstruction equality. Positivity of the bases gives the digit bounds and the next bound on the quotient. The loop variant is . At exit, forces , henceThere are exactly remainder operations and divisions, hence counted operations. The returned array occupies space and the remaining auxiliary space is constant. A represented value can never exceed , as follows either by the same nested expansion or by telescoping ; thus the stated range is also necessary for positive bases.
def intToDigs(n: Int, bs: Array[Int]): Array[Int] = {
// Pre: every base is positive, and 0 <= n < product(bs).
val N = bs.length
val xs = new Array[Int](N)
var i = 0
var q = n
while (i < N) {
xs(i) = q % bs(i)
q = q / bs(i)
i += 1
}
xs
}Place these methods inside the In the forward traversal, the loop invariant is that
LLBuffer class. The two private insertion methods operate on physical endpoints; the public methods interpret logical endpoints using the reversal flag.private def addFirst(x: Int): Unit = {
val q = new Node(x)
q.next = first
if (first == null) last = q else first.prev = q
first = q
size += 1
}
private def addLast(x: Int): Unit = {
val q = new Node(x)
q.prev = last
if (last == null) first = q else last.next = q
last = q
size += 1
}
def append(x: Int): Unit = {
if (backwards) addFirst(x) else addLast(x)
}
def prepend(x: Int): Unit = {
if (backwards) addLast(x) else addFirst(x)
}
def get(i: Int): Int = {
require(0 <= i && i < size)
val j = if (backwards) size - 1 - i else i
var q: Node = null
if (j <= size - 1 - j) {
q = first
var k = 0
while (k < j) { q = q.next; k += 1 }
} else {
q = last
var k = size - 1
while (k > j) { q = q.prev; k -= 1 }
}
val x = q.value
if (q.prev == null) first = q.next else q.prev.next = q.next
if (q.next == null) last = q.prev else q.next.prev = q.prev
q.prev = null
q.next = null
size -= 1
x
}
def rev: Unit = { backwards = !backwards }q is physical node , with ; its loop variant is . In the backward traversal, q is physical node , with ; its loop variant is . Thus both locate physical index , which is exactly logical index under the abstraction function.Deletion bypasses the chosen linked list node in both directions. Deleting the first or last linked list node updates the corresponding endpoint; deleting the only linked list node sets both endpoints to null. The surviving values keep their physical relative order, so the abstraction function gives precisely the required logical deletion. Each insertion preserves reciprocal links and adds its value at the physical end corresponding to the required logical end. Toggling the flag changes to its list reversal without moving any linked list nodes. ConsequentlyThese are worst-case time complexity bounds in the constant-cost pointer model; total storage is and each operation uses constant auxiliary space.
Use three-way quicksort. Fix the pivot value before moving any entries, and group equal entries together so that none needs a recursive call. These helpers and the sorter belong inside an enclosing scope containing The array partitioning loop invariant iswith unclassified. Swaps preserve the segment's array multiset and leave all entries outside it unchanged. The loop variant decreases in every branch. When swapping an entry from the right into
a.def swap(i: Int, j: Int): Unit = {
val t = a(i); a(i) = a(j); a(j) = t
}
def split(l: Int, r: Int): (Int, Int) = {
// Pre: 0 <= l < r <= a.length.
val p = a(l)
var lt = l
var i = l
var gt = r
while (i < gt) {
if (a(i) < p) {
swap(lt, i)
lt += 1
i += 1
} else if (a(i) > p) {
gt -= 1
swap(i, gt)
} else {
i += 1
}
}
(lt, gt)
}
def quicksort(l: Int, r: Int): Unit = {
require(0 <= l && l <= r && r <= a.length)
if (r - l > 1) {
val (lt, gt) = split(l, r)
quicksort(l, lt)
quicksort(gt, r)
}
}
// Sort the whole array with quicksort(0, a.length).i, do not increment i, since the incoming entry still needs classification. At termination,Here the middle segment is nonempty because the original pivot occurs somewhere in the preserved array multiset. Both recursive segments therefore have length strictly smaller than . Mathematical induction on segment length proves that the two recursive calls sort their segments, and their ordering relative to the middle segment proves the complete sorting postcondition.There is no additional array. The call stack and returned boundary pairs still use auxiliary space. An all-equal input takes one linear scan and no nontrivial recursive calls, so its time complexity is for . Balanced partitions give time complexity; a fixed first pivot can still give on adversarial distinct inputs. The worst-case call stack has entries.
Reuse A useful loop invariant is that the pending segments are pairwise disjoint valid array segments of length at least two; the whole array multiset equals its original value; and every inverted pair of positions lies in a single pending segment. Explicitly, if and , some queued contains both and . Thus all ordering obligations are localized to queued segments.
split and swap, but replace recursive calls with a FIFO queue of unresolved segments. Enqueue only segments of length at least two.def quicksortLoop(): Unit = {
val work = scala.collection.mutable.Queue[(Int, Int)]()
if (a.length > 1) work.enqueue((0, a.length))
while (!work.isEmpty) {
val (l, r) = work.dequeue()
val (lt, gt) = split(l, r)
if (lt - l > 1) work.enqueue((l, lt))
if (r - gt > 1) work.enqueue((gt, r))
}
}Initially the only possible pending segment is the whole array. One array partitioning confines all its remaining inversions to its two child segments. Its equal middle segment is ordered, the left/middle/right boundaries have the correct ordering, and segments of length at most one cannot contain an inversion. Because array partitioning only permutes the selected segment's entries, it cannot create an inversion across a boundary that previously had no inversions. This proves preservation of the loop invariant.
The loop variant is the total pending lengthProcessing one segment removes from and adds at most , strictly less because . Thus is a nonnegative integer and decreases. At exit no pending segment exists, so no inversion exists:The FIFO queue itself can grow to pairs. The time complexity is that of the corresponding recursive quicksort, including linear behavior on equal input. Removing recursion does not make the worst-case time complexity linear or eliminate the work storage.